JJ Abrams, who produced this film, has described this film as not a direct sequel to Cloverfield, but as a "blood relative" to the original movie. 10 Cloverfield Lane appears to be related to Cloverfield in name only, perhaps as an homage or as part of an anthology film series. No creatures, either "Clover" or its parasites, from that movie, appear in this film, nor are the events of Cloverfield referenced. It's also not a prequel to the original movie, as Michelle's iPhone comes from after 2008, and Emmett's bus ticket has a date from after 2008. The only name reference to Cloverfield comes from the title, which the end of the movie reveals is Howard's address. However, there are connections via the film's viral marketing. Howard appears as an employee of the month on the Tagruato "website", which is the parent company of Bold Futura, who owned the satellite that crashed into the sea and woke the Cloverfield monster up (the satellite can be seen crashing in the distance in the final scene in Cloverfield). In this movie, when Michelle enters the ventilation room, she finds an envelope from Bold Futura. Emmett also mentions that Howard worked for some big company that worked on "satellite stuff". It could be said that the Cloverfield incident was a distraction or test of Earth's defenses.
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No. J.J. Abrams once said, "if we used the same style again it may lose its gimmick; but we are not ruling out the fact that we may do it again." This meant that Cloverfield 2 might be filmed like a regular Hollywood movie to be able to tell a deeper story. The released movie confirms that the entire film is shot in traditional multi-camera style.
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Early on in the film, Michelle tries to escape the bunker, fearing that she has been kidnapped and lied to about a nuclear or chemical warfare attack. When she reaches the outer door of the bunker, a woman appears at the window with chemical burns all over her face, begging to be let in. Realizing there actually was an attack, Michelle retreats back into the bunker. The woman screams and curses at Michelle and begins to smash her head against the window. At the end of the film, when Michelle actually escapes the bunker, she sees the corpse of the woman, her flesh completely melted off. It's possible the woman killed herself by smashing her head against the window. It's also possible she simply succumbed to the original chemical attack and died.
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Howard confesses to Michelle that he was the one who crashed into her and he brought her back to his bunker for safety. As the movie progresses, Howard keeps mentioning his daughter, Megan and even shows her picture to Michelle, stating that her mother took her away from him (likely from his overbearing, controlling nature and his apocalypse paranoia...if his wife and daughter even existed). Howard then gives Michelle a box of clothes that belonged to Megan for her to wear. Emmett sees the picture later on and reveals that isn't Howard's daughter and that it was actually a girl from town that went missing. When Michelle reaches the top of the bunker, she finds a bloody earring and the word "HELP" scratched on the inside of the window as well as a photo of the same girl sitting next to Howard on his couch in the bunker, wearing the same earrings and the same clothes that Michelle is now wearing. So the interpretations vary, but are likely a combination of the following two. (1) Howard, being mentally unstable, missed his daughter so much that he kidnapped a girl that reminded him of her. She tried to escape, which angered Howard so he wound up killing her. Then abducted Michelle as a replacement, hoping she would comply due to a lack of choice, being the attack on the surface forced her to stay in the bunker. (2) Howard is a serial killer who never had a wife and daughter, he abducts young women who fit a certain description and keeps them captive in his bunker until he's ready to dispose of them. Hence the giant vat of acid he had handy. He needed to bide his time with Michelle however, due to Emmett forcing his way into the bunker.
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Spoilers

For a good portion of the movie, we as the audience are not entirely sure if Emmett is telling the truth about himself or if he is in on Howard's plan and just playing along. The bus ticket verifies that Emmett was telling the truth.
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